Grouping Genetic Algorithm with Efficient Data Structures for the University Course Timetabling Problem

Transcription

1 Grouping Genetic Algorithm with Efficient Data Structures for the University Course Timetabling Problem Felipe Arenales Santos Alexandre C. B. Delbem Keywords Grouping Genetic Algorithm Timetabling Problem 1 Introduction An efficient grouping genetic algorithm (GGA) was proposed by R. Lewis and B. Paechter(2007) to find feasible timetables for a version of the university course timetabling problem (UCTP). The algorithm proved to be efficient to construct feasible solutions for the UCTP benchmark instances (International Timetabling Competition 2002). In addition, the authors proposed a new set of harder instances, in order to better analyze their algorithm. In this paper, we propose an adequate data structure for the algorithm and preprocessing techniques that signficantly improve its efficiency. The organization of the paper is as follows. Section 2 defines the UCTP and presents the main characteristics of the GGA. Section 3 and 4 describes our contributions for the GGA applied to UCTP. Finally, Section 5 presents computational experiments and the conclusions. 2 The University Course Timetabling Problem A typical university timetabling problem consists of assigning a set u={u 1,...,u e} of e events (classes, exams, lectures, etc.) to t timeslots and r rooms in such a way as to satisfy a set of constraints and to optimize an objective function (R. Lewis and B. Felipe Arenales Santos University of Sao Paulo Sao Carlos, SP Brazil fearenales@grad.icmc.usp.br Alexandre Claudio Botazzo Delbem University of Sao Paulo Sao Carlos, SP Brazil acbd@icmc.usp.br 542

2 Paechter 2007). The constraints are classified into two types(burke et al. 1997):(i) hard constraints, which must be satisfied to a feasible timetable, and; (ii) soft constraints, which should be satisfied if possible. The hard constraints considered to this problem are: 1. Events requiring the same student should not be assigned to the same timeslot; 2. One room in one timeslot admits only one event; 3. All of the features required by an event should be satisfied by the room where the event is allocated, which has adequate capacity. The problem is NP-Complete (Garey and Johnson 1979), and the number of possible assignments grows exponentially with the input data. As a consequence, exact methods for UCTP are not proper for large instances. Therefore, heuristic-based methods have been widely investigated in the literature (Costa 1994; Abramson, Krishnamoorthy and Dang 1996; Thompson and Dowsland 1998; Schaerf 1999; Paechter et al. 1998; Socha and Samples 2003). Lewis and Paechter (2007) describe the applicability of a GGA to the UCTP, and propose a two-phase approach. In the first phase, the GGA improves the quality of the solutions (within a specified time) using a set of solution-builder heuristics, which speeds up the evolutionary process. In the second one, a local search algorithm does final improvements to promising solutions. However, as the input data increases too much (big cases of the proposed set), the convergence becomes slower and the first phase ends with high infeasibility level solutions. It should be worth noting that this paper deals with only the first phase and hard constraints. 3 Data Structures Some of the data structs used by the GGA are trivial, such as the set u, implemented as an array, and the timetable, represented by the r xt matrix T, where the element t ij is the event assigned to room i in timeslot j. Other matrices give the relation between students and events and between features and rooms. For further information, see Lewis and Paechter (2007). The most complex data structure envolved in this GGA, called feasibility matrix F, stores the places (one place is one room in one timeslot) where the events can be feasibly allocated. Its most efficient implementation (by the processing aspect) consists of a matrix whose rows correspond to places, and columns to events. It can be done by initially converting a two-dimensional matrix (G, rooms by timeslots), that contains the feasibility information of a given event, to a one-dimensional array of places H. Figure 1 depicts this step. g 11 g 1t G = H = ( ) ( ) g 11 g 1t g r1 g rt = h1 h p g r1 g rt Fig. 1: Linearization of the feasibility matrix to an event 543

3 Next, the resulting structure H is replicated e times (one for each event), resulting in the structure F. The F element f ij = 1 if event j can be feasibly assigned to a place i, and 0 otherwise, i = 1,...,p and j = 1,...,e, see Figure 2. F = ( f 11 f 1e H1 T ) HT e = f p1 f pe Fig. 2: Feasibility matrix construction In order to find out if one given event can be feasibly assigned to a place, no search is required, since the triple (event, room, timeslot) can be transformed into the pair (event, place) that corresponds to only one element of F. In this way, the cost to access this information is (in big-o notation) O(1) (Cormen et al. 2001; Knuth 1998). It speeds up the generation of solutions, i.e. the GGA can performe a larger number of generations for the same period of time compared to other relatively higher cost data structures, such as dynamic lists and other non-static data structures. Moreover, it is important to highlight that the larger the number of generations, the better the quality of solutions at the end of the GGA. 4 Preprocessing Techniques The values of the F elements are only related with the hard contraints satisfaction (see Section 2). Based on this, the hard constraints can be classified into two types: (i) static contraints, which define the assignment feasibility regardless the timetable state (i.e. the events allocated in it), as constraint 3; and (ii) dynamic constraints, which can change the feasibility of an assignment depending on the events already allocated in a timetable place, as constraints 1 and 2. Thus, the feasibility related to constraint 3 can be determined a priori. If an event requires a set of features and is required by n students,all the roomsthatdo nothaveall therequired featuresorhasthecapacityless than n can not be feasibly assigned to this event anytime. This information is stored in matrix A (event by rooms) that lists the event-rooms conflicts. Element a ij is 1 if the event i can not be assigned to the room j, i=1...e, j = 1,...,r; a ij = 0 otherwise. Thus, the values of F to a new solution can be obtained directly from A, avoiding unneeded processing. In addiction, the restriction number 1 can be modeled as a matrix B (event by event) that lists the event conflicts. The element b ij = 1 if the event i requires a student also required by the event j, i = 1,...,e, j = 1,...,e, i j; b ij = 0 otherwise. Observe that every iteration of a solution building inserts one unallocated event into the solution being built. 544

4 5 Computational Results The GGA and the proposed improvements was implemented in ANSI C under Linux (Ubuntu 8.04 distribution) on a computer with an Intel Pentium Dual-Core 1.73 GHz processor and 1 GB RAM. Considering the parameters as follow: timeslots number t = 45 (five days a week of five timeslots), mutation rate m r = 2, inversion rate i r = 4, recombination rate rr = 1.0, and population size ρ = 50. The input data is given by the instance sets. The experimental results was compared to the obtained by Lewis and Paechter (2007). The Figure 3 depicts the results obtained by the original implementation of the GGA and the implemented with improvements. The vertical axis corresponds to the distance to feasibility of the best solution present in the population. See Lewis and Paechter (2007) for further details about parameters and GGA implementation and run. (a) (b) (c) Lewis and Paechter (2007) Santos and Delbem (2010) Fig. 3: Behavior of the algorithm to (a) small, (b) medium, and (c) large instances. As can be noticed, the larger the input data, the greater the difference between the results obtained by the two implementations. It happens because as the input grows, the solution building time becomes significantly greater and the time saved is more expressive, showing the efficiency of the contributions proposed. 545

5 References 1. Abramson, D.; Krishnamoorthy, H.; Dang, H., Simulated Annealing Cooling Schedules for the School Timetabling Problem, Asia-Pacific Journal of Operational Research, vol. 16, pp (1996). 2. Burke, E. K.; Kingston, J.; Jackson, K.; et al., Automated university timetabling: the state of the art, The Computer Journal, vol. 40(9), pp (1997). 3. Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C., Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill (2001). 4. Costa, D., A tabu search algorithm for computing an operational timetable, European Journal of Operational Research, vol. 76, pp (1994). 5. Garey, M. R.; Johnson, D. S., Computers and intractability: a guide to the theory of NPcompleteness. Freeman and Company, New York (1979). 6. Knuth, D., The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, pp (1997). 7. Lewis, R. and Paechter, B., Finding Feasible Timetables Using Group-Based Operators, IEEE Transactions on Evolutionary Computation, vol. 11(3), pp (2007). 8. Paechter, B.; Rankin, R.; Cumming A.; Fogarty, T., Timetabling the Classes of an Entire University with an Evolutionary Algorithm, in Parallel Problem Solving from Nature (PPSN) V (Lecture Notes in Computer Science, vol. 1498), Baeck T.; Eiben A.; Schoenauer M.; Schwefel, H., Eds. Berlin, Germany: Springer-Verlag, pp (1998). 9. Socha, K. and Samples, M., Ant Algorithms for the University Course Timetabling Problem with Regard to the State-of-the-Art, in Evolutionary Computation in Combinatorial Optimization (EVOCop) III (Lecture Notes in Computer Science vol. 2611), Dorigo, M.; Di Caro, G.; Sampels, M., Eds. Berlin, Germany: Springer-Verlag, pp (2003). 10. Thompson, J. M. and Dowsland, K. A., A Robust Simulated Annealing based Examination Timetabling System, Computers and Operations Research, vol. 25, pp (1998). 546